Different expressions for an integral if I split the region of integration

Question

Consider the assumptions

$Assumptions = 1 > v > 0 && y1 \[Element] Reals && y2 \[Element] Reals &&   k > 0 && 1 > t1 > 0 

and define

y[t_, y1_, y2_, t1_] = 
      Piecewise[{{y1, 0 <= t <= t1}, {y2, t1 < t <= 1}}, 0];

and two different formulations of the same integral,

Assuming[t1 + v < 1, 
  Integrate[
   Assuming[{0 <= t <= 1, t1 + v < 1}, 
    Integrate[(1 + 
        Exp[-2 Integrate[y[s, y1, y2, t1]^2, {s, t - u - v, t - v}]])/
      2, {u, 0, t}]], {t, v, t1 + v}]] // Simplify

and

Assuming[t1 + v < 1, 
   Integrate[
    Assuming[{0 <= t <= 1, t1 + v < 1}, 
     Integrate[(1 + 
         Exp[-2 Integrate[
            y[s, y1, y2, t1]^2, {s, t - u - v, t - v}]])/2, {u, 0, 
       t - v}]], {t, v, t1 + v}]] + 
  Assuming[t1 + v < 1, 
   Integrate[
    Assuming[{0 <= t <= 1, t1 + v < 1}, 
     Integrate[(1 + 
         Exp[-2 Integrate[
            y[s, y1, y2, t1]^2, {s, t - u - v, t - v}]])/2, {u, t - v,
        t}]], {t, v, t1 + v}]] // Simplify

The two formulations differ for the region of integration, that in the second case is splitted in two parts, because the interval in u between 0 and t is splitted in [0,t-v]+[t-v,t],and then the integrals summed. Why Mathematica give me back two different expressions?

Answer 1

Maybe not the answer you're looking for. It took a long time to run your code as it is so I changed it a bit. The bottom line is that Mathematica is doing fine and the two expressions you get just look different.

Some definitions :

myAssum = 1 > v > 0 && y1 \[Element] Reals && y2 \[Element] Reals && 1 > t1 > 0
myConditions = {tMv -> t - v};

because until the last integration everything depends on t-v only, so no point carrying two variables.

intY2 = Simplify[Integrate[PiecewiseExpand[y[s, y1, y2, t1]^2], {s, tMv - u, tMv}, 
                 Assumptions -> {0 <= tMv <= t1, 0 <= u <= t}], 
        Assumptions -> {myAssum, 0 <= tMv <= t1, 0 <= u <= t}];

intExp = Integrate[(1 + Exp[-2 intY2])/2, {u, 0, t},  
                   Assumptions -> {0 <= tMv <= t1, 0 <= t <= 2}];

Now we substitute back tMv -> t-v and do the last integration :

result = Integrate[intExp /. myConditions, {t, v, t1 + v}, 
                   Assumptions -> myAssum];

To check the other approach I just repeat the last two steps :

intExp1 = Integrate[(1 + Exp[-2 intY2])/2, {u, 0, tMv},  
                    Assumptions -> {0 <= tMv <= t1, 0 <= t <= 2}];
intExp2 = Integrate[(1 + Exp[-2 intY2])/2, {u, tMv, t},  
                    Assumptions -> {0 <= tMv <= t1, 0 <= t <= 2}];

result1 = Integrate[intExp1 /. myConditions, {t, v, t1 + v}, 
                    Assumptions -> myAssum];
result2 = Integrate[intExp2 /. myConditions, {t, v, t1 + v}, 
                    Assumptions -> myAssum];

Check :

(result == result1 + result2 ) // Simplify
(* True *)